integral x+1×dx/x^2-1 using integration by parts
Answers
Answer:
Answer:
Given :-
The nth term of an AP is 3, 8, 13, 18 is 73.
To Find :-
What is the value of n or number of terms of an AP.
Formula Used :-
\clubsuit♣ General term or nth term of an AP Formula :
\begin{gathered}\bigstar \: \: \sf\boxed{\bold{\pink{a_n =\: a + (n - 1)d}}}\: \: \: \bigstar\\\end{gathered}
★
a
n
=a+(n−1)d
★
where,
\sf a_na
n
= nth term of an AP
a = First term of an AP
n = Number of terms of an AP
d = Common Difference of an AP
Solution :-
First, we have to find the common difference (d) :-
Given :
a₁ = 3
a₂ = 8
According to the question :
\begin{gathered}\implies \bf Common\: Difference(d) =\: a_2 - a_1\\\end{gathered}
⟹CommonDifference(d)=a
2
−a
1
\implies \sf Common\: Difference(d) =\: 8 - 3⟹CommonDifference(d)=8−3
\implies \sf\bold{\purple{Common\: Difference(d) =\: 5}}⟹CommonDifference(d)=5
Hence, the common difference or d is 5 .
Now, we have to find the value of n :
Given :
nth term \sf (a_n)(a
n
) = 73
First term (a) = 3
Common Difference (d) = 5
According to the question by using the formula we get,
\dashrightarrow \bf a_n =\: a + (n - 1)d⇢a
n
=a+(n−1)d
\dashrightarrow \sf 73 =\: 3 + (n - 1)5⇢73=3+(n−1)5
\dashrightarrow \sf 73 - 3 =\: (n - 1)5⇢73−3=(n−1)5
\dashrightarrow \sf 70 =\: (n - 1)5⇢70=(n−1)5
\dashrightarrow \sf \dfrac{\cancel{70}}{\cancel{5}} =\: (n - 1)⇢
5
70
=(n−1)
\dashrightarrow \sf \dfrac{14}{1} =\: (n - 1)⇢
1
14
=(n−1)
\dashrightarrow \sf 14 =\: n - 1⇢14=n−1
\dashrightarrow \sf - n =\: - 1 - 14⇢−n=−1−14
\dashrightarrow \sf {\cancel{-}} n =\: {\cancel{-}} 15⇢
−
n=
−
15
\dashrightarrow \sf\bold{\red{n =\: 15}}⇢n=15
\therefore∴ The value of n or number of terms of an AP is 15 .
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EXTRA IMPORTANT FORMULA :-
\diamond⋄ General term or nth term of an AP Formula :
\begin{gathered}\bigstar \: \: \sf\boxed{\bold{\pink{a_n =\: a + (n - 1)d}}}\: \: \: \bigstar\\\end{gathered}
★
a
n
=a+(n−1)d
★
where,
\sf a_na
n
= nth term of an AP
a = First term of an AP
n = Number of terms of an AP
d = Common Difference of an AP
\diamond⋄ Sum of nth term of an AP Formula :
\begin{gathered}\bigstar \: \: \sf\boxed{\bold{\pink{S_n =\: \dfrac{n}{2}\bigg\lgroup 2a + (n - 1)d\bigg\rgroup }}}\: \: \: \bigstar\\\end{gathered}
★
S
n
=
2
n
⎩
⎪
⎪
⎪
⎧
2a+(n−1)d
⎭
⎪
⎪
⎪
⎫
★
where,
\sf S_nS
n
= Sum of nth term of an AP
n = Number of terms of an AP
a = First term of an AP
d = Common Difference of an AP